TECHNICAL FIELD OF THE INVENTION
[0001] The present invention generally relates to target ranging methods and systems and
more particularly to a covert ranging method and system based on passive target ranging
in conjunction with active sensing that fixes target range to resolve range and speed
ambiguity.
BACKGROUND OF THE INVENTION
[0002] For a wide variety of military and civilian aviation scenarios air-to-air passive
ranging provides a useful way to determine distances between aircraft. The goal of
air-to-air passive ranging is to determine the range from one aircraft, called "ownship,"
to another, called "target", by detecting energy emanating from the target. For this
purpose, "ownship" may be equipped with directional receivers that measure the angle
of the arriving energy during a data collection interval. To this are added fixes
of "ownship's" position and heading obtained from an inertial navigation system (INS)
on-board ownship. For most applications, the energy will be in the radio-frequency
(RF) or infrared (IR) regions of the electromagnetic spectrum.
[0003] A RF system may monitor radar transmissions and RF communications from the target
for these purposes. Very long detection ranges are feasible that substantially exceed
those at which target could acquire ownship on its own radar. Thus, ownship may be
able to locate target without itself being detectable. In this situation, the primary
advantage of passive ranging is gained: stealth. Long acquisition range also affords
ownship more time to detect target's presence. The directional receiver may be implemented
by a two-axis RF interferometer or several such units covering different fields of
view.
[0004] The infrared system would sense black body radiation emanated from the target. Therefore,
operation does not depend on target transmission protocol. Detection range is considerably
reduced and ownship would be within range of target's radar. Two factors mitigate
this risk: (1) the target might restrict radar usage to avoid detection and (2) convergence
time is considerably less for IR passive ranging systems. The latter is attributable
to the shorter operating ranges and the higher spatial resolution of the IR sensor.
The directional receiver may be implemented using a FLIR imager and video tracker.
[0005] A principal deficiency in all air-to-air passive ranging method is ill-conditioning.
This is a condition in which small errors in the measurements can cause much larger
ones in the computed ranges. There are two principal causes of ill-conditioning: (1)
a limited baseline for ranging data; and (2) the need to infer target motion from
the data. The relation between baseline ranging accuracy may be explained in terms
of triangulation. This is appropriate, although there is no explicit triangulation
step in the ranging algorithm, since a triangulation principle is at work whenever
range is estimated from sightings at different locations. Factors affecting triangulation
accuracy will have similar effects on ranging accuracy. Two of these are the lengths
and direction of the baseline ranging data.
[0006] Accordingly, there is a need for a method and system that avoids ill-conditioning
in passive air-to-air ranging by overcoming the limited ranging data baseline and
target motion inference problems of the prior art.
SUMMARY OF THE INVENTION
[0007] The present invention, accordingly, provides a covert ranging method and system that
overcomes or reduces disadvantages and limitations associated with prior passive air-to-air
ranging methods and system.
[0008] One aspect of the invention is a method for covertly determining and predicting air-to-air
target range and speed data relative to a predetermined position. The method includes
the steps of passively sensing a target to produce a passive target data set and then
relating the passive target data set to a predetermined position. This produces a
transformed passive data set that may be compared to a predicted target data set.
This comparison generates a measurement error. The method further actively senses
a target for a minimally detectable period of time to produce an active target data
set and then relates the active target data set to the previously determined measurement
error to produce a system error. Then, the method changes the predicted target data
in response to the system error. In an operational target environment, the method
and system of the present invention repeat the above steps continuously to covertly
determine target data relative to the predetermined position.
[0009] A technical advantage of the present invention is that by using random or quasi-random
flashes of active radar signals, the method and system perform covert ranging and,
at the same time, overcome the limited ranging data base line and target motion inference
problems associated with a purely passive ranging system.
[0010] Another technical advantage of the present invention is the simplifying of computations
necessary for dynamic modeling of the air-to-air passive ranging problem. By bracketing
the scope of the search necessary for ranging, the present invention improves the
computational efficiency of the air-to-air ranging algorithm. In the present invention,
penalties are imposed when target data values exceed predetermined brackets. Additionally,
optimal magnitudes of dynamic target model perturbations are determined that assure
a level of stability within the dynamic target model. This avoids the detrimental
effects of ill-conditioning in the air-to-air passive ranging algorithm.
BRIEF DESCRIPTION OF THE DRAWINGS
[0011] The invention and its modes of use and advantages are best understood by reference
to the following description of illustrative embodiments when read in conjunction
with the accompanying drawings, wherein:
FIGURE 1 provides a flow block diagram of the covert ranging system of the preferred
embodiment;
FIGURE 2 shows factors effecting passive ranging accuracy;
FIGURE 3 illustrates the geometry of air-to-air ranging;
FIGUREs 4 and 5 illustrate the operation of optimizing the dynamic ranging model according
to the preferred embodiment;
FIGURE 6 shows two orders of special penalty that the preferred embodiment uses; and
FIGURE 7 illustrates the effects of bracketing the solution of the preferred embodiment.
DETAILED DESCRIPTION OF THE INVENTION
[0012] The preferred embodiment of the present invention is best understood by referring
to the FIGUREs wherein like numerals are used for like and corresponding parts of
the various drawings.
[0013] The covert ranging system of the preferred embodiment is based on a passive ranging
system involving the use of an active radar sensor. The active radar obtains a fix
on the target range by a "flash illumination" a transmission too short to detect in
most circumstances. Consequently, ownship's location is not compromised in any circumstances.
This active radar fix resolves the range and speed ambiguity that may otherwise slow
conversion of the completely passive ranging system. Once resolved, the passive ranging
system maintains an accurate fix on range as long as the target maintains at constant
velocity. Any subsequent departure from the assumed flight path can be detected and
corrections provided by a subsequent "flash illumination."
[0014] FIGURE 1 shows a system block diagram 10 of the covert ranging system of the preferred
embodiment. Beginning with passive sensors 12, covert ranging system 10 shows that
bearings relative to the body axes of the sensing aircraft are taken and sent to coordinate
transformation unit 14. At coordinate transformation unit 14, navigation data from
the aircraft's inertial navigation system (INS) 16 transforms these bearings into
azimuth and elevation data relative to the INS 16 coordinate system. While actual
passive data is being received and transformed, a resident model in an associated
computer system generates data as indicated by dynamic target model block 18. The
resident model outputs this data as indicated by block 20 to provide predict bearings
from model. At junction 22, actual data including bearings having azimuth and elevation
are joined and compared with the predicted bearings from the dynamic target model
to generate a measurement error. Then, at junction 24, computed penalties from block
26 are input to junction 24. These computed penalties take into consideration the
output of constraint functions from block 28, as well as INS data from block 16 and
parameters of the resident dynamic target model of block 18. Other outputs from the
constraint functions block 28 include an input to block 30, discussed below. Constraint
functions block 28 receives an output from active radar portion 38. Active radar portion
38 receives an active sensor input 40, a trigger signal indicated by block 42, and
pointing information 36, as already mentioned.
[0015] Inertial navigation system 16 provides data to seven places in the covert ranging
system 10 of the preferred embodiment. These include coordinate transformation block
14, predicted bearing block 20, compute penalty block 26, compute direction of model
perturbation block 30, determine limit magnitude of perturbation block 32, pointing
information block 36, and operator display unit 50. Similarly, dynamic target model
18 provides data to six points including predict bearings from model block 20, compute
penalties block 26, compute direction of model perturbation block 30, determine limit
of magnitude of perturbation block 32, pointing information block 36, and display
to operator block 50.
[0016] Returning to junction 24, the composite of the output from computer penalties block
26 and the measurement error from junction 22 is formed to comprise a "system error.'
The system error goes to compute direction of model perturbation block 30, which gives
the maximum reduction of system error. Output from model perturbation direction block
30 combines with output from perturbation magnitude limit block 32 to produce at block
44 an optimum magnitude of perturbation. This is performed by a "linear search" method
such as Davidon's algorithm as described in "Davidon's Cubic Interpolation Method,"
Methods of Optimization, G.R. Walsh, ed. Sec. 39, pp. 97-102 (1975). From block 44, an optimal model perturbation
results and a signal goes along line 46 to the resident dynamic target model within
the associated computer system for operating the model at block 18.
[0017] In an operational system, for example, as well as the covert ranging system 10 continuously
updating the target data, output goes to a monitor such as monitor 48 that display
to operator block 50 to exhibit a variety of useful information concerning target
and ownship, for example, a running display of target's position relative to ownship.
Also, operator control 52 may activate trigger signal block 42 to cause active radar
38 to send a flash of active radar energy that active sensors 40 receive. This may
be in response to a signal from display to operator block 50 that an active fix would
be beneficial, as when target acceleration.
[0018] In the preferred embodiment, constraint functions block 28 uses constraints such
as location at time, speed, maximum range, etc., that may be switched on or off according
to the availability of necessary and relevant data. Additionally, active radar block
38 may instead be a laser device or "ladar" that generates a laser signal in the light
or infrared energy producing wavelengths of the electromagnetic spectrum.
[0019] The following discussion illustrates the passive air-to-air ranging problem that
the preferred embodiment solves. This material may be supplemented by unpublished
Appendix A for a more complete understanding. In air-to-ground ranging, for example,
the target may be presumed stationary and ranging amounts to determining its special
coordinates in three-dimensional space. Thus, an air-to-ground model representing
the target has three degrees of freedom. This may be further reduced to two degrees
of freedom if the altitude of ownship relative to target is known. The situation is
more complicated in air-to-air ranging where the model must describe target's position
as a function of time. Accordingly, the air-to-air ranging problem applies a model
known as "kinematic" model. The additional degrees of freedom necessary for the kinematic
model aggravate ill-conditioning, because they decrease the redundancy through which
noise is rejected. To recover from the ill-conditioning that occurs, the data collection
interval must be extended. This has the effect of slowing conversions of the algorithm.
[0020] In the passive air-to-air ranging problem, the form of the kinematic model is assumed
a priori. This, in effect, limits the scope of the motion that may described. Over a short
interval, a simple model suffices. The more complex motion possible in a longer interval
may be accompanied by a more complex model. However, this may be counterproductive
because of the associated need for more data and a longer data collection interval.
Therefore, the preferred embodiment adopts a relatively simple model having six degrees
of freedom. This is equivalent to assuming that the target flies in approximately
a straight line path during data collection. For reliable performance, the convergence
time of the algorithm should be short enough to make this a good assumption. Therefore,
factors affecting rate of convergence, such as measurement accuracy, are extremely
important. However, it has been shown that un-modeled target accelerations are not
always detrimental to ranging accuracy.
[0021] FIGURE 2 shows a conceptual data flow diagram 60 to illustrate the interplay of factors
affecting ranging accuracy in the preferred embodiment. With reference to FIGURE 2,
data flow diagram 60 has at its center ranging accuracy domain 62. Ranging accuracy
domain 62 is affected by the flight path geometries that domain 64 indicates (i.e.,
data collection internal) sensor accuracy that domain 66 indicates, time that domain
68 indicates and INS accuracy that domain 70 indicates.
[0022] An example of the interplay for ranging accuracy that domain 62 indicates may be
as follows. Ranging accuracy 62 may be maintained while the data collection internal
68 is reduced by improving sensor accuracy 66 or as a result of more favorable flight
path geometries 64. Favorable geometries are characterized by high angular rates of
change of bearings. As a result, a shorter data collection internal is possible at
shorter ranges (e.g., it takes twice as long to establish a particular geometry at
a 100 km as at 50 km). A system which requires a shorter data collection interval
to attain a given accuracy is said to "converge faster."
[0023] FIGURE 3 graphically depicts an air-to-air encounter 80 that the preferred embodiment
addresses. Consider air-to-air encounter sketch 80 where ownship 82 seeks to obtain
a range to target 84. In this encounter, coordinate system 86 moves in translation
so that ownship 82 remains at the origin 88 at all times. The directions of the axis,
X₁, X₂, and X₃ are then fixed in inertial space by references from the ownship INS.
Bearing measurements may then be assumed as referenced to these axis rather than to
the body axis of ownship. Time is taken as zero at the onset of data collection. In
this scenario, an important limitation is that range is indeterminate in the absence
of ownship acceleration. As a result, there is a fundamental ambiguity between range
and speed. This means that there are infinitely many target models consistent with
a particular set of sightings in such a case. As described in Appendix A, the preferred
embodiment solves this ambiguity problem.
[0024] The preferred embodiment characterizes the formulation of passive ranging as an inverse
problem. The goal of the inverse problem is to find the kinematic model which is most
consistent with the measured azimuth and elevations recorded during the data collection
interval. These measurements are referred to as "actual data." Consistency is determined
by numerically computed "synthetic data" from the model in comparing on a one-two-one
basis with the actual data. The result is summarized by a single non-negative number
called "measurement error." As mentioned, the form of the model is assumed
a priori. The task of the algorithm therefore is to evaluate the parameters of the model.
U.S. Patent Application Serial No. 07/008,432 (TI-11782A), entitled "Method and Apparatus
for Air-to-Air Aircraft Ranging" by Choate et. al. and assigned to Texas Instrument
Incorporated describes this process and is here incorporated by reference.
[0025] As described in U.S. Patent Application Serial No. 07/008,432 (TI-11782A), the minimization
problem uses the following recursive procedure to dynamically model target data:
(a) a start-up model is chosen;
(b) a perturbation of the model δm, which causes the measurement error, J, to decrease is computed;
(c) a search is conducted along a straight line in the parameter space of the model
for the minimum of J. The model which minimizes J becomes the new resident model.
The direction of a straight line is given by (b);
(d) if several consecutive iterations leave the model virtually unchanged, the resident
model is taken as the solution. Otherwise, a new iteration is begun at (b).
[0026] To find the perturbation direction as indicated at block 30 of FIGURE 1, the preferred
embodiment derives approximate expressions for
which is accurate for small δ
m and which is simple enough that a closed form expression can be found for δ
. This solved for δ
. Only the direction is used, magnitude is recomputed in step (c), above. The approximate
expression is a Taylor series truncated after the second or third term which may be
derived by several associated solution techniques.
[0027] One solution technique to this problem may be the gradient method, also known as
the method of steepest descent. Another method may be the conjugate gradient method.
Yet another method may be Newton-Raphson algorithm, which utilizes a second degree
approximation that remains accurate for probations of much greater magnitude. The
Newton-Raphson algorithm makes feasible larger step sizes with fewer iterations to
achieve conversions. This process, unfortunately, requires increased per-iteration
processing. Another method that the preferred embodiment may use is a computation
of the Hessian matrix used in the Newton-Raphan algorithm. All of these solution techniques
are described in more detail in unpublished Appendix A.
[0028] Perturbation magnitude is determined in step (c) of the ranging algorithm by a "linear
search." This terminology derives from the fact that the minimum sighting error is
sought along a straight line path in the parameter space of the model. The line passes
through the origin and is directed in the perturbation direction determined in step
(b) above. The only free parameter is perturbation magnitude, therefore, the search
is one-dimensional. The minimum is determined by trial-and-error using a Davidon's
Cubic Interpolation Method as stated previously. The computations are based on the
exact model, rather than an approximation of finite degree, and, therefore, are themselves
exact. This is the advantage of using step (c) to determine perturbation magnitude.
[0029] Davidon's algorithm works as follows in the preferred embodiment. A bracket containing
the minimum is established. Normally, this is done by increasing the upper limit of
the bracket until either of two conditions are meet:
(1) the derivative of J in the search direction is positive; or
(2) J is larger than at the origin.
However, using the constraints introduced later herein, the preferred embodiment
may compute it directly, and, thus avoid trial-and-error in most cases. Next, J and
its first order derivative in the direction of the search are computed at the boundaries
of the bracket. These four numbers determine the cubic polynomial which is taken as
a model of J within the bracket. The minimum of the cubic may be readily computed
analytically. The location of the minimum becomes a new boundary. Whether "upper"
or "lower" depends on the value of the derivative of J. The following Table 1 defines
this relationship:
TABLE 1
Sign of Derivative of J |
Classification of boundary |
Negative |
Lower |
Positive |
Upper |
If the derivative is zero, the location is returned as the solution of the linear
search. Since the new boundary divides the original bracket, the width of the new
bracket is reduced. If "small enough", the center of the bracket is returned as the
solution of the linear search. Otherwise, a new polynomial is fitted to the new bracket
and the procedure iterated. The current program exits when the uncertainty in range
is 100 meters or less, and the uncertainty in velocity is 0.1 meters per second, or
less.
[0030] FIGUREs 4 and 5 illustrate these operations. The solid curves give J as a function
of perturbation magnitude. In FIGURE 4, the initial bracket is the integral bracket
[0,4]. The dashed curve is a plot of cubic polynomial matched to J and its derivative
at the end-points. The fit in the interior is not particular good, but the minima
occur at similar locations. It is found that J is increasing at the minima of the
cubic polynomial β = 1.975. Therefore, this becomes a new upper boundary. The polynomial
has been refitted [0,1.975] in FIGURE 5 [dashed curve]. The approximation is now quite
accurate and the minima occur at virtually identical locations. This illustrates the
rapid convergence that typifies the Davidon algorithm. However, in later iterations
of the ranging algorithm, the topology of J can be exceptionally flat and subject
to artifacts caused by numerical noise. As a result, a newly defined boundary may
be consistently upper (lower) and situated very near the previous one. The preferred
embodiment includes "accelerators" that detect condition and force a minimum reduction
of bracket width.
[0031] The preferred embodiment employs constraints for making available to the ranging
algorithm knowledge of the real-world that is not explicitly present in the data.
The constraint function, by restricting the domain of feasible solutions, excludes
unrealistic target models. This makes conversions more rapid and alleviates ill-conditioning
to a large degree. While additional computations are needed to implement the constraint
functions, these are largely offset by a reduction in the number of iterations. By
using the constraints to determine the initial bracket in Davidon's algorithm, numerical
overflow (underflow), which sometimes occurs when evaluating constraint functions
within the stop-band, is avoided. The action of the constraints is indirect. More
specifically, they are responsive to functions of model parameters--not to the parameters
themselves. For example, the speed constraint is responsive to the magnitude of target
velocity. Therefore, changes in velocity which preserve speed have no effect on the
constraint. Further, the speed constraint is independent of the position components
of the model.
[0032] The constraints are implemented as penalty functions which return non-negative numbers
(penalties) which are very small when the constraint is satisfied and large when it
is violated. The penalties add to the measurement error to yield to the "system error,"
which is minimized in solving the ranging problem. The fundamental structure of the
ranging algorithm is unaffected by the constraints. Only the numbers appearing in
the equation change. The additive nature of the constraints allows them to be developed
and tested independently--a property of considerable practical importance.
[0033] The cost of implementing the constraints is substantially reduced by their independence
of the measurements (time samples). Thus, calculations do not have to be performed
over a sequence of N time samples, as do computations relating to measurement error.
Further, a penalty function is often dependent on only a subset of the model parameters.
[0034] Since the penalties are negligibly small when the constraints are satisfied, the
constraints have little effect on the result when the model is well within the feasible
solution space. Clearly, it is important to choose constraint parameters so as not
exclude the true target model. This must be balanced with the desire to restrict the
feasible solution space for better ranging performance. To optimize this trade-off,
it may be attractive to adapt the constraint parameters for each individual target.
This could employ
a priori information derived from intelligence sources or inferred from an electronic characterization
of target transmissions.
[0035] FIGURE 6 illustrates two orders, denoted by the symbol M, for speed penalty. Note
that as order increases, the influence of the constraint within the bandpass is reduced.
In the limit (M→∞), the penalty function takes on the shape of rectangular well with
zero penalty within the bandpass and infinite penalty outside. This is ideal theoretical
behavior. But, from a practical standpoint orders above four or eight do not offer
significant advantages.
[0036] Another useful constraint is a limit on maximum target acquisition range. This is
known approximately from consideration of the power of target's radar, antenna gain
of ownship's receiver, ambient noise level, etc. The location of target at acquisition
is a basic component of the kinematic model of the target (see
x(0), equation (2) below). Range is simply the Euclidian norm of
x(0). Any exponential power of range, typically a positive even integer, can be used as
the penalty function. The penalty is weighted to realize a desired penalty at the
specified maximum acquisition range.
[0037] The preferred embodiment further includes a flight path adviser for processing early
sighting data to do the following:
(1) Recommend an ownship maneuver to optimize ranging performance;
(2) Determine the start-up model to initiate data inversion; and
(3) Detect a change of target velocity (i.e., target acceleration) indicating the
need to revise the target model and possibly, a new active radar fix.
"Early sighting data" refer to sightings collected at the beginning of the data collection
interval, for which ownship's velocity may be taken as approximately constant. Under
this condition, relative target motion is confined to a plane in three-dimensional
space. This implies that the salient features of the motion may be described in only
two dimensions. As noted earlier, range can not be resolved in the absence of ownship
acceleration. However, in the preferred embodiment it is possible to determine range
if target speed is known. This is not the case, but often a reasonable estimate of
speed is available. An accurate start-up model can be determined by "flash ranging."
[0038] From the flight path adviser, it is possible to determine noise level to indicate
sensor performance. It is also possible to indicate the efficacy of the flight path
geometry during a sensing. Appendix A illustrates how each of these steps may be performed.
In the preferred embodiment, flash ranging significantly improves the accuracy of
ranging algorithm. In particular, by including with a preferred embodiment the flash
ranging using active sensors 40 of FIGURE 1, serious ill-conditioning problems are
overcome. To more fully understand the flash ranging aspect of the preferred embodiment,
the following discussion provides a mathematical derivation of its use in the ranging
algorithm. Additionally, Appendix B provides an exemplary source code listing of the
ranging algorithm of the preferred embodiment including the active flash ranging aspect.
Dealing particularly with the flash ranging aspect of the preferred embodiment, we
begin an exemplary mission at time,
t = 0 and then at sometime later, for example at
active sensors 40 turn on to obtain a range fix on the target. For purposes of covert
operation, the use of active sensors 40 is random or quasi-random. Active sensors
40 are off most of the time, thus making them very difficult to detect so that the
target cannot sight or range on ownship. There are several kinds of information that
can be determined from the active sensor:
(1) Range
(2) Doppler
(3) Bearing (azimuth and elevation).
Combined, range and bearing determine target's location relative to ownship. This
can advantageously be used as a constraint when bearings from the active sensor are
more reliable than those from the passive sensor. However, the passive sensor is at
no fundamental disadvantage for determining bearing and may be preferred for this
purpose. In this case, range information alone would be utilized for the constraint.
Doppler gives the component of relative target velocity along the line-of-sight. This
information is particularly difficult to determine passively and thus is highly complementary.
As a constraint it would be applied to the velocity component of the model.
[0039] In the following we will describe how active range information can be incorporated
as a constraint. Techniques for constructing constraints from active doppler and bearing
measurements are similar. The range fix that active sensors 40 and active radar 38
obtain may be denoted, ρ
_{N}, meaning the nominal range at time τ. Since the use of active radar is expected to
be very short and occasionally employed, in discussing the preferred embodiment, the
use of active sensors 40 will be denoted "flash ranging." Note that flash ranging
would be of limited value if it were not possible to maintain an accurate estimate
of target location by passive means the majority of the time. As stated previously,
the method and system of the preferred embodiment assume a linear kinematic model
for relative target motion. This model may have the following model expression:
It will also be convenient to express the model as the following expression:
where the matrix L
(t) takes the following form:
and
m is a column vector consisting of the parameters of the model as follows:
The estimate ρ
_{N} at τ is introduced into the passive ranging method and system as a constraint that
is implemented through the penalty functions that block 26 computes. The penalty function
for the computations of block 26 may take the form:
where
K_{γ} is the "gain", Ω is the "bandwidth," and ρ is the range computed from the model expression
for
x(t) of Equation (3) at time τ through the following relationship:
In Equation (7), dependence on the time parameter τ is implicit. The exponent "2"
appearing in (6) is used for the purpose of illustration and may be replaced by other
positive even integers. For convenience in the following derivations, assume that
This multiplicative scale factor may then be restored to the result.
[0040] In the air-to-air passive ranging algorithm, by introducing constraints through penalty
functions and then adding them to the cost function of the system, it is possible
to determine how well the measurements fit the measurements and the allowed scope
of the model.
[0041] By computing partial derivatives of γ with respect to model parameters, the following
expressions obtain:
and
With these expressions, it is possible to develop expressions for the derivatives
of range with respect to model parameters. From Equation (7),
As a result,
where
e_{i} is a unit vector of dimension six. It follows from Equation (3) that
where
l_{i}^{τ} is the
i^{th} row of
L^{τ}.
[0042] By computing second derivatives,
Defining the 6x6 matrix as
and
Further, denoting the gradient of γ with respect to the model parameters
m as
the following expression results from Equations (8) and (12).
where
Also, in modifying the ranging algorithm of United States Patent Application Serial
No. 07/008,342, as described above, it is important to modify the Hessian matrix that
has the following definition:
The elements of the Hessian matrix were determined in Equation (9) above. However,
a simpler expression results by substituting from Equation (15) to derive the following
expression:
It will be recalled, that the scale factor
was assumed equal to 1 with the expectation that it would be restored subsequent to
computation. This is possible now by simply multiplying
H by
Moreover, numerical computations are greatly simplified by noting that
and
with
and
for i=1, j= 1,3. Note that the derivative expressions now only need to be evaluated
for three elements of the six elements
g, and six elements of the 36 elements of
H.
[0043] The gradient
g and the Hessian
H are used to approximate the range-at-time penalty function γ. An exact definition
of γ is given by equation (6). The approximation is given by
and holds for all δ
m of small magnitude. Used within the Newton-Raphson procedure, (a) can be used to
compute in closed form the perturbation δ
m for which
m+δ
m yields the minimum γ (within the limits of the approximation). However, the objective
is to minimize the system error J, not just the component γ. The extension presents
no formal difficulty because system error is the sum of measurement error plus all
penalties. The approximation for system error is simply the sum of expressions of
the form (a), each of which may be derived independently. The approximate nature of
the truncated series representation (a) limits accuracy of the solution δ
m when the magnitude δ
m is not very small. As a result, it is usually possible to improve the solution by
rescaling it to optimum magnitude. This is done through a "linear search," so named
because the path of the search is a straight line in solution space. There is no approximation
error in the linear search, although the path is usually suboptimal.
[0044] In one-dimensional optimization problems of this sort, it is possible to search in
a way similar to that described in U.S. Patent Application Serial No. 07/008,342.
However, constraints offer the opportunity to further bracket the scope of the search
and, therefore, improve the computational efficiency of the search algorithm. As a
result, it is possible to examine how the brackets are determined. For this purpose,
the preferred embodiment assumes that, at the outset of the search, the system error
is J₀. Obviously, then a successful search cannot result in a penalty exceeding J₀
(i.e., it is known that
in the preferred embodiment). The bracket boundaries are defined such that equality
holds in Equation (25), thus yielding
Where α² is
and the carets " ̂" an "
" denote values at the bracket boundaries. Thus, bracket boundaries are attained when
Since ρ is not the free parameter of the search, consider now its relationship to
this parameter, denoted β.
During the search, the dynamic target model may be expressed in the following form
where β is the "distance" in the direction of the search δ
m, and
m₀ is the unperturbed model. The corresponding relative target position is given by
the following expression:
or
with
Taking the square of the Euclidian norm of Equation (30),
Solving for β, it is then possible to obtain the expression
Because only the searches in the positive δ
m direction are allowed, negative solutions to Equation (33) or those that are not
purely real, are not accepted in the algorithm of the preferred embodiment.
[0045] FIGURE 7 illustrates the bounding that occurs to obtain values for β̂₁ and β̂₂ by
substituting ρ̂ (See Equation (27)) for ρ and Equation 33. Values for β̂₁ and β̂₂
are determined from ρ̂ in a similar manner. Referring to FIGURE 7, there is shown
the effect of the constraint for the penalty function. With curve 110 representing
the penalty function the example imposes the constraint (27) so that the penalty cannot
exceed J₀ without leaving the bracket [
, ρ̂]. Additionally, with this constraint, curve 112 shows Case A where no solution
occurs for Equation (33), since no positive real β yields a ρ within the bracket.
Curve 114 shows Case B where a solution only at ρ̂ occurs. Curve 116 shows Case C
where a solution occurs at ρ̂ and
. By straight forward differential calculus, it can be shown from Equation (32) that
If this exceeds ρ̂², the square of the largest ρ for which γ equals J₀, there clearly
is no solution. This is Case A that curve 114 describes. If,
The solutions β̂₁ and β̂₂ exist for ρ̂, but there are no solutions for
. Curve 114 shows this as Case B which is bracketed by [β̂₁,β̂₂]. This is that portion
118 to the left of the ρ̂ dash line 120. If, further,
there are solutions β̂₁, β̂₂ for ρ̂ and solutions
₁,
₂ for
. Curve 116 shows this as Case C as bracketed by [β̂₁,
₁] and [β̂₂,
₂] as depicted by those portions 122 and 124 between dash line 120 for ρ̂ and dash
line 126 for
. Thus, the linear search must consider two ranges in Case C which curve 116 passes.
On the other hand, if a speed constraint or maximum acquisition range is imposed,
a narrower range for the optimal solution may be established.
[0046] It should be noted that the maximum range constraint may be absorbed into the range-at-time
of constraint once ρ
_{N} is established by the flash ranging active radar fix. As usual, the bandwidth Ω is
geared to the precision to which ρ
_{N} is known and is small when ρ
_{N} is believed to be accurate.
[0047] Although the above description adequately illustrates the operation of the preferred
embodiment, unpublished Appendix B as stated previously provides a source code listing
to more explicitly illustrate a functioning example the operation of the "flash ranging"
as well as conversion aspects of the preferred embodiment.
1. A method for covertly determining and predicting target data relative to a predetermined
position, comprising the steps of:
(a) passively sensing the target to produce a passive target data set;
(b) relating said passive target data set to a predetermined position to produce a
transformed passive data set;
(c) comparing said transformed passive data set to a predicted target data set to
generate a measurement error;
(d) actively sensing a target for a minimally detectable period of time to produce
an active target data set;
(e) relating said active target data set to said measurement error to produce a system
error;
(f) changing said predicted target data set in response to the said system error.
2. The method of Claim 1, further comprising the step of repeating steps (a) through
(f) to covertly determine target data continuously relative to said predetermined
position.
3. The method of Claim 1, wherein said passively sensing step further comprises the step
of generating azimuth and elevation data relative to an ownship sensor.
4. The method of Claim 3, wherein said passive target data set relating step further
comprises the step of relating said azimuth and elevation data to an inertial navigation
system position.
5. The method of Claim 1, further comprising the step of constraining said active target
data set by a plurality of penalty functions upon determining that certain aspect
of said active target data set exceeds a predetermined limit.
6. The method of Claim 5, further comprising the step of using said penalty functions
to associate small weights with said active target data set when a predetermined aspect
of said active target set fails within said predetermined limit and associates large
weights with said active target data set when said predetermined aspect falls outside
said predetermined limit.
7. The method of Claim 5, wherein said penalties comprise a speed penalty associated
with a speed measurement aspect of said active target data set.
8. The method of Claim 5, wherein said penalties further comprise a maximum acquisition
range penalty associated with a maximum acquisition range aspect of said active target
data set.
9. The method of Claim 1, wherein said predicted target data set comprises predicted
bearing data, said predicted bearing data comprising predicted azimuth data and predicted
elevational data.
10. The method of Claim 1, further comprising the step of limiting said changing step
by a perturbation magnitude, said perturbation magnitude associated with said system
error.
11. The method of Claim 10, further comprising the step of limiting said perturbation
magnitude using a linear search procedure.
12. The method of Claim 11, wherein said linear search procedure comprises a cubic interpolation
method for limiting said perturbation magnitude.
13. The method of Claim 1, further comprising the step of limiting said changing step
by a perturbation direction, said perturbation direction associated with said system
error.
14. The method of Claim 1, wherein said passive target data set comprises early sighting
data for advising of a flight path of the target.
15. A system for covertly determining and predicting target data relative to a predetermined
position, comprising:
a passive sensor for passively sensing the target to produce a passive target data
set;
coordinate transformation circuitry for relating said passive target data set to
a predetermined position to produce a transformed passive data set;
comparison circuitry for comparing said transformed passive data set to a predicted
target data set and thereby generating a measurement error;
an active sensor for actively sensing a target for a minimally detectable period
of time to produce an active target data set;
system error circuitry for relating said active target data set to said measurement
error to produce a system error; and
perturbation circuitry for changing said predetermined target data set in response
to said system error.
16. The system of Claim 15, further comprising circuitry for iteratively changing said
predicted target data set in response to said system error and thereby covertly determining
target data continuously relative to said predetermined position.
17. The system of Claim 15, further comprising circuitry for determining azimuth and elevation
data of the target relative to said passive sensor.
18. The system of Claim 15, further comprising circuitry for relating passive sensor azimuth
and elevation data to a set of data from an inertial navigation system.
19. The system of Claim 15, further comprising constraint functions circuitry for constraining
said active target data set by a predetermined set of penalty functions.
20. The system of Claim 19, wherein said penalty functions impose small penalties for
data within predetermined limits and substantially larger penalties for data within
said active target data set outside said predetermined limits.
21. The system of Claim 20, wherein said penalty functions comprise a speed penalty function.
22. The system of Claim 20, wherein said penalty functions comprise a maximum acquisition
range penalty function.
23. The system of Claim 15, wherein said predicted target data set comprises predicted
target bearing data, said predicted target bearing data further comprising predicted
target azimuth data and predicted target elevation data.
24. The system of Claim 15, wherein said perturbation circuitry further comprises circuitry
for determining perturbation magnitude and perturbation direction.
25. The system of Claim 24, wherein said perturbation magnitude determining circuitry
comprises circuitry for performing a linear search using a cubic interpolation method.